I would like to estimate the conditional scale function $(\sigma_\tau(X_t))$ in a QAR-QARCH model represented by: \begin{equation} Y_t = \mu_\tau(X_t) + \sigma_\tau(X_t)\epsilon_t,\, t = 1,2,\ldots \end{equation} where $\epsilon$ is zero (0) $\tau-$quantile \begin{equation} P(\epsilon_t\leq0) = \tau \end{equation} and unit (1) scale, \begin{equation} P(\rho_\tau(\epsilon_t)\leq1) = \tau \end{equation} with $\rho_\tau(a) = a(\tau - I(a\leq0))$ is called the check-function. Using the definition of first probability, we can write: \begin{equation} P(Y_t \leq \mu_\tau(X_t)\mid X_t) = \tau = F_{X_t}(\mu_\tau(X_t)) = E[I(Y_t \leq \mu_\tau(X_t))\mid X_t] \end{equation} and \begin{equation} P(\rho_\tau\left(Y_t - \mu_\tau(X_t)\right)\leq\sigma_\tau(X_t)\mid X_t) = \tau \end{equation} The non-parametric estimation of $\sigma_\tau$ depends on the difference $Y_t-\mu_\tau(X_t)$ but my problem is that they don't have the same length because $\mu$ also is estimated by the use of bins (with NW for instance). What could be the alternative?
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